Complex Input
in [DSP]
|
From: glen herrmannsfeldt on 14 Nov 2009 10:58 Phil O. Sopher <invalid(a)invalid.invalid> wrote: (snip) > We use Complex Numbers to represent sinusoids (should that be > "cisoids"?) that have both a magnitude and also a phase relationship. > Why do we do this? > Well, we don't have to, because we can do everything with sin(wt+a) > and cos(wt+a), but when we differentiate, we change from sin to cos > and sometimes the sign changes (cos to sin) and sometimes it > doesn't (sin to cos). What a mess of rules to remember and be > inherently error-prone. > However, if you wish, you can work in those terms with what > I assume you will consider to be real input. In this case, complex values are a convenience over hard to remember and work with trig identities. The complex values are not physical, but convenient. > If we could represent everything by an exponential, e^wt, then the > differential is always w.e^wt which is easier to deal with. > It all stems from e^jwt = cos(wt) + j.sin(wt). There is, however, another place that complex numbers can be used, and that is in the exponent. That is, if we have both sinusoids and exponential decay (or growth) then a complex exponent represents both. This comes directly from the solution of some second order differential equations. While you still take the real part at the end, it is not the real part of the complex quantity in the exponent, but of the result of exp(). In this case, it doesn't seem so strange to consider the value in the exponent as a complex physical quantity. Dielectric constant and the related index of refraction are some that have this property. -- glen
From: John Monro on 14 Nov 2009 20:58 Jerry Avins wrote: > John Monro wrote: >> jasonkee111 wrote: >>> Hi. i read a lot of article focusing on real input. In physical world, >>> does the input in complex form? If exist, where it comes from and >>> what is >>> the application? Or actually it is just theory only? >>> >>> Thanks >>> >>> >> >> Hi Jasonkee111, >> >> Actual, physical signals in complex form do exist and are commonly >> available at the back of high-class communications receivers. > > No. What is available there are two real signals, one representing I, > the other Q. Jerry, I am sure you are very familiar with the origins of the terms 'real' and 'imaginary' but if you want to have a bit of fun with the terms then that is OK with me. I did think though that the OP deserved a seious answer, hence my posting. As the OP may not be as familiar with DSP as you, I will make a few points: The word 'imaginary' is used in common language to describe something that is purely the product of the mind and does not have a physical form. In contrast, the word 'real' describes something that does have a physical form. In DSP the words have a completely different meaning. The term 'imaginary' is used to describe one component of a complex signal and the term 'real' is used to describe the other component. In Communications Engineering they are caled I and Q. As used in the technical sense 'real' and 'imaginary' are not opposites at all and have only a historical connection to the common language use of those words. > >> These receivers have two actual BNC connectors marked I (in-phase) and >> Q (quadrature phase). For each sinusoidal component in the I signal >> there is a corresponding equal-amplitude sinusoidal component present >> in the Q signal, but offset in phase by 90 degrees. >> >> Having a complex signal available makes subsequent analog or digital >> signal processing simpler by suppressing the negative-frequency >> component. >> >> With on negative frequencies present signal can be shifted up or down >> in frequency without the complication of having a 'image' signal being >> produced. >> >> It is an interesting contradiction that the signals with the simplest >> spectrum(positive frequencies or negative frequencies only) require >> two signal wires (I and Q) while more complicated signals with >> mirror-image positive and negative-frequency components require only >> one signal wire (plus earth in all cases). > > This is an excellent example of mathematical simplicity masking reality. All of DSP can be explained using nothing more than Euclid, but do we really want that level of reality all the time? The OP might comment here. Jasonkee111, did you find my posting useful to you? > Tell me, John, Which leg of a two-phase 220VAC power line gives > imaginary shocks? > > Jerry Neither, in the case as stated. In this case the two legs have a phase relationship of 180 degrees, so no imaginary component is available. If, hypothetically, you had a four-phase system then you could identify two legs that have a phase relationship of 90 degrees to each another. You could designate these the 'real' phase and the 'imaginary' phase. If I contacted that 'imaginary' leg I would expect to get an 'imaginary' shock. (1) Note 1. As this is the DSP group all terms are used in the technical sense, so an 'imaginary' shock is one in which the contraction of my muscles is offset by 90 degrees relative to they way they would contract if I received a shock from the 'real' leg.) Regards, John
From: Jerry Avins on 14 Nov 2009 21:22 John Monro wrote: > Jerry Avins wrote: >> John Monro wrote: >>> jasonkee111 wrote: >>>> Hi. i read a lot of article focusing on real input. In physical >>>> world, >>>> does the input in complex form? If exist, where it comes from and >>>> what is >>>> the application? Or actually it is just theory only? >>>> >>>> Thanks >>>> >>>> >>> >>> Hi Jasonkee111, >>> >>> Actual, physical signals in complex form do exist and are commonly >>> available at the back of high-class communications receivers. >> >> No. What is available there are two real signals, one representing I, >> the other Q. > > Jerry, > I am sure you are very familiar with the origins of the terms 'real' and > 'imaginary' but if you want to have a bit of fun with the terms then > that is OK with me. The only bit of fun came later. > I did think though that the OP deserved a seious answer, hence my > posting. As the OP may not be as familiar with DSP as you, I will make > a few points: My answer was intended to be serious. Perhaps it was too terse. > The word 'imaginary' is used in common language to describe something > that is purely the product of the mind and does not have a physical > form. In contrast, the word 'real' describes something that does have a > physical form. > > In DSP the words have a completely different meaning. The term > 'imaginary' is used to describe one component of a complex signal and > the term 'real' is used to describe the other component. In > Communications Engineering they are caled I and Q. > > As used in the technical sense 'real' and 'imaginary' are not opposites > at all and have only a historical connection to the common language use > of those words. > > >> >>> These receivers have two actual BNC connectors marked I (in-phase) >>> and Q (quadrature phase). For each sinusoidal component in the I >>> signal there is a corresponding equal-amplitude sinusoidal component >>> present in the Q signal, but offset in phase by 90 degrees. >>> >>> Having a complex signal available makes subsequent analog or digital >>> signal processing simpler by suppressing the negative-frequency >>> component. >>> >>> With on negative frequencies present signal can be shifted up or down >>> in frequency without the complication of having a 'image' signal >>> being produced. >>> >>> It is an interesting contradiction that the signals with the simplest >>> spectrum(positive frequencies or negative frequencies only) require >>> two signal wires (I and Q) while more complicated signals with >>> mirror-image positive and negative-frequency components require only >>> one signal wire (plus earth in all cases). >> >> This is an excellent example of mathematical simplicity masking reality. > > All of DSP can be explained using nothing more than Euclid, but do we > really want that level of reality all the time? Of course not. Nevertheless, I think it is important to have that in mind from the outset. > The OP might comment here. Jasonkee111, did you find my posting useful > to you? > >> Tell me, John, Which leg of a two-phase 220VAC power line gives >> imaginary shocks? This is my sortie into whimsy. > > Neither, in the case as stated. In this case the two legs have a phase > relationship of 180 degrees, so no imaginary component is available. Actually, no. Technically, center-tapped 240 is not two phase. Real two-phase power was phased out (pun unavoidable) in the years following WWII, but power companies supplied it to users who needed it with a Scott-T transformer connection from a three-phase line. (Used the other way, a Scott T will convert two phase to three. A pair of Scott Ts can be used instead of open deltas to save iron. http://en.wikipedia.org/wiki/Scott-T_transformer > If, hypothetically, you had a four-phase system then you could identify > two legs that have a phase relationship of 90 degrees to each another. > You could designate these the 'real' phase and the 'imaginary' phase. > If I contacted that 'imaginary' leg I would expect to get an 'imaginary' > shock. (1) > > Note 1. As this is the DSP group all terms are used in the technical > sense, so an 'imaginary' shock is one in which the contraction of my > muscles is offset by 90 degrees relative to they way they would contract > if I received a shock from the 'real' leg.) Now that gives a real shock in the arm! Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
From: John Monro on 14 Nov 2009 22:32 Jerry Avins wrote: (snip) >> >>> Tell me, John, Which leg of a two-phase 220VAC power line gives >>> imaginary shocks? > > This is my sortie into whimsy. >> >> Neither, in the case as stated. In this case the two legs have a >> phase relationship of 180 degrees, so no imaginary component is >> available. > > Actually, no. Technically, center-tapped 240 is not two phase. Real > two-phase power was phased out (pun unavoidable) in the years following > WWII, but power companies supplied it to users who needed it with a > Scott-T transformer connection from a three-phase line. (Used the other > way, a Scott T will convert two phase to three. A pair of Scott Ts can > be used instead of open deltas to save iron. > http://en.wikipedia.org/wiki/Scott-T_transformer > >> If, hypothetically, you had a four-phase system then you could >> identify two legs that have a phase relationship of 90 degrees to each >> another. You could designate these the 'real' phase and the >> 'imaginary' phase. >> If I contacted that 'imaginary' leg I would expect to get an >> 'imaginary' shock. (1) >> >> Note 1. As this is the DSP group all terms are used in the technical >> sense, so an 'imaginary' shock is one in which the contraction of my >> muscles is offset by 90 degrees relative to they way they would >> contract if I received a shock from the 'real' leg.) > > Now that gives a real shock in the arm! > > Jerry Thanks for that Jerry. I had never heard of the Scott-T transformer. The reference was interesting, particularly the way you could see the phase relationship clearly from the circuit diagram alone. No need for a phasor diagram here! Regards, John
From: Jerry Avins on 15 Nov 2009 16:50
John Monro wrote: ... > Thanks for that Jerry. I had never heard of the Scott-T transformer. > The reference was interesting, particularly the way you could see the > phase relationship clearly from the circuit diagram alone. No need for > a phasor diagram here! There are many attributes of polyphase power distribution that widen one's horizon. The meeting of that and DSP lies mostly in line- and motor-protection "relays". (In power circles, a relay is a sensor gizmo designed to trip a circuit breaker.) I know that many see me as the curmudgeon of comp.dsp who claims that complex numbers, while useful for computation, don't embody reality. Whatever complex numbers embody, they make it easy for a neophyte to imagine unreality. The abstraction they represent allows one to skip altogether the abstracted concepts. That happens in other areas too. In a recent thread, a newbie wondered how to convert PCM to integer. You must be aware of my being asked by a CS graduate how computer hardware distinguishes between ASCII and bite-wide integers, and blowing me off in disbelief when explained. (I repeat it often enough.) You may also be aware of Richard Feynman's disappointment by that the Brazilian graduate student he was interviewing. The student knew plane-parallel refraction theoretically, but couldn't think of an instance of it, not even a window pane. Whatever the mathematics, if you don't know how it works, you don't know what you're talking _about_, quantum mechanics excepted. An example from polyphase power, which is where we started. Balanced three-phase systems are fairly easy to deal with. (More on this later.*) Unbalanced systems, those in which the phase currents differ (and with faults, also the phase voltages) are more complicated. One way to regularize the subject is decomposing the into "symmetrical components" (http://en.wikipedia.org/wiki/Symmetrical_components) which consist of positive (A B C) and negative sequence (A C B) balanced currents, and zero sequence currents which have the same magnitude and phase in all lines. When their coefficients are appropriately chosen, the superposition of these components represents the actual currents. The abstraction is so useful that some people always deal with three-phase circuits this way. "It can be shown" that two wattmeters suffice to measure the total power delivered by a three-phase system. A standard three-meter installation in a wye-connected system consists of a current coil in each line ans the corresponding voltage coil from line to neutral; basically, three single-phase meters. It gets more interesting in a delta-connected system because there is no neutral. If the voltage coils have the same impedance, connecting them together and letting the node float to create a virtual neutral works. What happens if the impedances differ? Analysis with line phasors is straightforward. With symmetrical components, it is long and difficult. Yet I have seen it done that way because symmetrical components are "real" and the actual line phasors are "derived". Bah! The upshot of the analysis is that, so long as one cares only about the total power, it doesn't matter what the node voltage is. The node can be tied to one of the lines, effectively shorting out one of the meters. I have a simple proof of that case which makes no use of symmetrical components and little use of line phasors. Connect the current coils and one side of the corresponding voltage coils to lines A and B, and the other ends of the voltage coils to line C. Connect a load from A to C. one meter correctly indicated the power in that line. Likewise for a load from B to C. Connect a load from A to B. Two current coils sense it, while the voltage coils see a voltage 60 degrees out of phase. The power is then 2*I*V(cos(60) = VI. Power is additive: Q.E.D It is important in engineering to deal with what is there and not scratch one's left ear with the right hand behind one's back. I chose examples peripheral to DSP in the hope that newness will allow a fresh view. Jerry ________________________________ * Long ago, in order to get credit for a Fourier analysis course I had taken in another school, I had to pass an evaluation test. It turned out to be a regular class exam about 3/4 of the way through the semester. It consisted of one question: "A balanced three-phase wye-connected system delivers square waves to a balanced resistive load. Describe the neutral current." (Three-phase circuits were covered in an prerequisite course.) The key word was "describe". I first did it graphically. At any instant, two phases have the same polarity and the other is opposite, so the magnitude of the neutral current is the same as any of the phases. Every time any phase undergoes a zero crossing, the neutral polarity reverses, so the neutral current has three times the frequency of the line currents. Period. End of story. Still, engineering is math, so I figured I better supply an equation as well as a graph. I turned the page in the exam book and wrote out the series for a square wave, describing one phase. (You don't derive the quadratic formula every time you need it. Why should a square wave be different?) I didn't time-shift it to get the other two phases and go through the trig to combine them. Instead, I invoked a three-phase theorem (from the prerequisite; it was legit): in a wye system, harmonics divisible by three add in the neutral, the others cancel. So I put a 3 in front of the 4/pi, and crossed out the appropriate terms. Then, to ice the cake -- I already knew the answer, I took the 3 inside the brackets, canceling one factor of three in all the denominators and replaced 3f with f'. Voila! a square wave in f'. Total elapsed time, about 6 minutes. One other fellow got the right answer. He was a math whiz. He derived the square wave, time shifted it, added, canceling some terms and tripling the rest but omitted the neat f' substitution, and so didn't visualize the actual wave shape. A few others nearly made it but ran out of time (50 minutes.) When I turned in the exam book, the prof sympathetically told me to try, that I didn't have to be perfect. I whispered that I thought I had completed it. He looked, wrinkling his nose at the graph. It was clearly correct, but not really what he wanted. I whispered again, "There's more. Turn the page." He did, and in a second, gave me a thumbs up. -- Engineering is the art of making what you want from things you can get. ����������������������������������������������������������������������� |